Analytical study of integrating downhole thermoelectric power generation with a coaxial borehole heat exchanger in geothermal wells

Geothermal power generation employing Organic Rankine Cycle (ORC) technology is a widely acknowledged and conventional approach for harnessing geothermal energy. In an innovative advancement, we propose a novel design integrating downhole thermoelectric power generation with a coaxial borehole heat exchanger. This design aims to enhance the efficiency and sustainability of geothermal energy utilization. In this innovative design, the geothermal well is divided into two distinct sections: a power generation section and a heat exchanging section, achieved through the implementation of a packer positioned from the uppermost part of the targeted zone. The process involves the injection of cold fluid downhole via an insulated pipe. Subsequently, a portion of the injected fluid is directed to flow in reverse within the casing-tubing annulus above the packer, while another portion circulates into the casing-tubing annulus below the packer before ascending through the tubing. This dual flow mechanism establishes distinct cold and hot sources for the thermoelectric generator, a key feature facilitated by this innovative design. Analytical models detailing of downhole temperature distribution for thermoelectric power have been meticulously developed. A comprehensive case study, focusing on a geothermal well with 3000 m length of power generation section and 500 m heat exchanging section, has been conducted. The results indicate that a significant generating capacity could be achieved with a higher wellhead temperature, and the payback period under different carbon tax scenarios is about 6–8 year. Furthermore, the effects of injection rate, fluid diversion ratio, and casing-tubing configuration on power performance and thermal-electricity efficiency are also discussed. This method not only enables the concurrent harvesting of geothermal energy and power generation but also operates consistently throughout the year. The results thus emphasize the viability and economic feasibility of the proposed approach.

compact and high in power density, which is preferable for the application of medium-or high-temperature power generation.Kim et al. 26 investigated the waste heat recovery performance of a direct contact thermoelectric generator on a diesel engine.The results showed that a 10 K decrease in the coolant temperature yields a smaller increase (0.25%) in the conversion efficiency for the engine, while a 20 K decrease causes a larger increase (15%) in the conversion efficiency.However, these attempts are all harvesting heat to generate electricity on a smaller scale, and very limited attention has been paid to generating electricity in downhole with more than thousands of feet length.
The aim of this work is to find an alternative method in which electric power may be generated in the downhole with thermoelectric technology and coaxial borehole heat exchanger technology in geothermal wells, which is different from traditional ORC or Kalina technology.The coaxial borehole heat exchanger recovers heat from the surrounding formation, and transfers the heat to thermoelectric modules to generate power in the downhole.Mathematical models are developed to study the downhole temperature distributions related to power generation.Then, a case study with cost-benefit analysis will be conducted to validate the potential of downhole thermoelectric power generation in geothermal wells under different carbon taxes.Finally, the factors that affect the performance of downhole thermoelectric power generation are investigated, and the optimum operational parameters are determined.

Thermoelectric generator
A thermoelectric generator (TEG) is a device that directly converts a heat gradient into electricity through the Seebeck effect, which is the phenomenon that electric current is induced by a temperature difference in an electrical conductor.A TEG is usually made up of ceramic substrates, electrical insulators, electrical conductors, and many N-type and P-type semiconductors with high Seebeck cofficients that are alternately connected in series electrically and in parallel thermally, as shown in Fig. 1.The Seebeck coefficient is the generated voltage difference per degree of temperature difference of a material.It is material-and temperature-dependent, with negative values for the N-type and positive values for the P-type semiconductors.
Electrical resistivity and thermal conductivity are two other important parameters when selecting materials for thermoelectric power generation and maintaining a high heat gradient.Seebeck coefficient, electrical resistivity, and thermal conductivity are combined into the figure of merit, Z , which is a measure of the heat-to-electricity conversion efficiency.Usually, the dimensionless figure of merit,ZT , which is the product of the figure of merit by the average temperature, is used to compare the properties of materials.The higher the ZT is, the higher the heat-to-electricity conversion efficiency will be [27][28][29] .
For a thermoelectric pair with cross-areas of A N and A P for N-type and P-type semiconductors, with the same length of L , the thermal conductance K , and electrical resistance R , could be expressed as 27,30 , where, K n and K p are the thermal conductance of N-type and P-type semiconductors, respectively, W/K; k n and k p are the thermal conductivity of N-type and P-type semiconductors, respectively, W/(m K); R n and R p are the electric resistance of N-type and P-type elements in a TEG, respectively, Ω; σ n and σ p are the electrical resistivity of N-type and P-type semiconductors, respectively, Ω m; A n and A p are the cross-section areas for N-type and P-type semiconductors, m 2 ; L np is the length of semiconductors, m.
The figure of merit (Z) is defined as, (1) where, k is the thermal conductivity of a pair of N-type and P-type semiconductors, W/(m K); σ is the electrical resistivity of a pair of N-type and P-type semiconductors, Ω m; and α is the overall Seebeck coefficients of a pair of N-type and P-type semiconductors, V/K.

System design of downhole thermoelectric power generation system with coaxial borehole heat exchanger
A Downhole coaxial heat exchanger is a way to extract geothermal energy without producing geothermal fluid 11,13,31,32 .The extracted heat from the working fluid is usually used for direct utilization or ORC power generation.Recently, power generation with thermoelectric technology has been proposed to harvest and utilize geothermal energy in downholes 22,[33][34][35][36] .Combining with a coaxial borehole heat exchanger, a new design of downhole thermoelectric power generation in geothermal wells is presented (Fig. 2).
In the proposed design, the well bottom is sealed completely, and the casing is lowered to the well bottom to prevent leakage between the formation and borehole.Tubing is run into casing and downed to right above the sealed bottom.A packer will be set at the top of production zone to isolate the casing-tubing annulus.An injection pipe (vaccum-insulated steel pipe 37 ) assembled with a downhole fluid diverter will be run into the casingtubing annulus and just pass through the packer.The downhole fluid diverter is located just above the packer and is used to adjust fluid flowing into the casing-tubing annulus above and below the packer.The outer surfaces of the tubing above the packer are fully covered with thermoelectric modules.Cold fluid will be continuously injected downward in the injection pipe from the surface.When the cold fluid flows pass the downhole fluid diverter, part of the cold fluid will be diverted and enters into the casing-tubing annulus above the packer; the other part will continuously flow downwards through the packer and enters the casing-tubing annulus below the packer, which will reverse back at the circulation point due to the sealed bottomhole.The diverted cold fluid at the downhole fluid diverter flows upwards along the casing-tubing annulus above the packer and provides cold sources for TEGs.While the other part of the cold fluid continuously flows downwards along the casingtubing annulus below the packer, it adsorbs heat from the surrounding formation gradually and becomes hightemperature fluid at the bottom of the well.At the circulation point, the heated fluid flows upwards in tubing to the surface and provides heat sources for TEGs.At the same well depth, the temperature of upward flowing fluid in tubing is higher than that of upward flowing fluid in casing-tubing annulus.Once the system achieves a stable temperature difference between the tubing and casing-tubing annulus by continuously injecting and circulating the cold fluid, electricity will be generated as a response to the applied temperature gradient, and the produced electricity could be transmitted to the surface and input to the local grid.In general, the proposed design includes two sections: one is the heat harvesting section located below the packer, which harvests heat from the surrounding hot formation with a coaxial borehole heat exchanger; and the other is the thermoelectric power generation section located above the packer, which produces power by measuring the temperature difference across both sides of TEGs.In order to create a temperature difference as large as possible across both sides of TEGs, the upper casing above the packer is coated with insulation materials (such as nano-SiO 2 aerogel 37 ) to reduce heat-transfer from the surrounding formation to the fluid flowing in casing-tubing annulus, while the lower casing below the packer has a good heat conductivity to harvest enough geothermal energy from the deeper formation to heat the injected cold fluid in it.The tubing and injection pipe surfaces are also coated with insulation materials.
Compared to TEG applications in other industries, downhole power generation represents a large-scale application, especially given the depth of geothermal wells.Suzuki and Tanaka 38 pointed out that the output power has a decreasing variation after first increasing with the increased length of the TEGs.Montecucco et al. 39 disclosed that connecting thermoelectric generators in series is better for electrical system efficiency than in parallel when the temperature differences remain constant.Therefore, segmented TEGs are used and connected in series when considering the longer depth of the well.

Downhole temperature model
To assess the performance of the proposed downhole power generation design, it is necessary to know the temperature distributions along the tubing, the casing-tubing annulus, and both sides of TEGs.The assumptions are made: (1) Injection pipe is perfectly insulated; the packer length and the injection pipe length extending out of the packer are ignored that is, the inlet temperatures of the injected cold fluid at the casing-tubing annulus above the packer and below the packer are the same as the inlet temperature at surface; (2) The heat transfer between formation and wellbore is in a steady state; (3) Temperature drops across both tubing and casing walls are neglected due to the high thermal conductivity of metals as well as the small thickness of the walls; (4) Thermoelectric elements in TEG are identical, and their geometric configurations are in the optimum form; (5) External heat-transfer irreversibility between the thermoelectric devices and the heat reservoirs are neglected; (6) Seebeck coefficient, thermal conductance, electrical resistance, and figure of merit of the thermoelectric devices are independent of temperature in the range of studied temperatures; (7) Fluid in tubular flows in onedimension axial direction and heat conducts in one-dimension radical direction.
Heat exchanges among tubing fluid, casing-tubing annular fluid, coaxial borehole heat exchanger, and surrounding formation result in temperature differences on both sides of TEGs.To obtain temperature distributions in the casing-tubing annulus, tubing, and coaxial borehole heat exchanger, a heat balance over an element of length, dz , which is treated as a control volume at a distance of dz from the surface where z equals to zero, was built.Here z is positive in the downward direction.The schematic heat balance for the tubular and formation is depicted in Fig. 3.

Below packer (heat harvesting section)
The flow conduits below the packer work as a coaxial borehole heat exchanger.In the casing-tubing annulus below the packer, the injected fluid is in direct contact with the casing inner wall.Assuming the casing is cemented with the rock in good condition, the heat transfer between the rock and the casing happens by conduction, and between the casing wall and the reversed fluid in casing-tubing annulus by convection.The convection into the formation is ignored.In the tubing, the reversed fluid from the circulation point flows up and enters the thermoelectric power generation section; the heat transfer occurs only through the tubing wall 12,40,41 .In this study, water was selected as the working fluid.For the fluid flows from depth of z to (z + dz) in casing-tubing annulus and from depth of (z + dz) to z in tubing below the packer, energy balance equations could be established accordingly.where q fhbp is the heat of the reversed fluid in tubing below the packer, J; q fcbp the heat of the injected fluid in casing-tubing annulus below the packer, J; q tabp is the heat flow from the tubing to the casing-tubing annulus, J; q Fbp is the heat flow from the surrounding formation to the casing-tubing annulus, J.
Based on the heat-transfer theory, the heat balance for fluid flowing in the tubing and in the casing-tubing annulus below the packer is respectively given by: where c w is the water specific heat capacity, J/(kg.K); w injbp is the mass of the injected fluid in the casing-tubing annulus below the packer, kg; T fhbp is the fluid temperature in the tubing below the packer, K; T fcbp is the fluid temperature in the casing-tubing annulus below the packer, K.
Heat flow, q Fbp , from surrounding formation to the casing-tubing annulus below the packer is given below, Heat flow, q ta , from tubing to casing-tubing annulus below the packer is given by: where r c is the casing radius, m; r t is the tubing radius, m; T wb is the temperature at wellbore/formation interface, K; U abp is the overall heat-transfer coefficient of casing-tubing annulus below the packer, which depends on the resistances to heat flow through casing-tubing annular fluid, casing metal, and cement, W/(m K); U tbp is the overall heat-transfer coefficient of tubing the packer, which depends on the resistances to heat flow through the tubing fluid and tubing metal, W/(m K).U abp and U tbp can be calculated by many methods 42,43 .
Assuming that the temperature at the wellbore/formation interface along the vertical direction changes linearly, that is: where, T surface is the surface temperature of the wellbore/formation interface, K; g G is the geothermal gradient, K/m; z is the well depth from surface, m.
Letting, Simplifying these equations based on the assumptions of incompressible, single-phase fluid, and the following equations can be obtained, The boundary conditions could be found to be that the temperature of the reversed fluid at the tubing inlet is equal to the temperature of the injected fluid in the casing-tubing annulus at the circulation point, and the temperature of the injected fluid at the outlet of the injection pipe is known.Here, we assume that the depth of the tubing inlet (circulation point) is the same as the well bottom.Circulation point at the tubing inlet and injection pipe outlet is expressed as: where L is the depth of the tubing inlet or the circulation point or the well bottom, m; L c is the depth of cold fluid entering into the casing-tubing annulus below the packer or the depth of the downhole fluid diverter or the depth of injection pipe outlet, m; T inj is the temperature of injected liquid at the surface, K.

Above packer (power generation section)
For the fluid heated in the casing-tubing annulus below the packer and flowing upward in the tubing, it enters at the depth of (z + dz) and leaves at z with heat convection towards the casing-tubing annulus above the packer; and for the fluid flowing upward in the casing-tubing annulus above the packer, the energy balance involves heat transfer from the tubing to the casing-tubing annulus above the packer and heat-transfer from surrounding formation 40,41 .Therefore, energy balance equations could be established in tubing and casing-tubing annulus, accordingly.
where q fh is the heat of the reversed fluid in tubing above the packer, J; q fc the heat of the reversed fluid in casingtubing annulus above the packer, J; q ta is the heat flow from tubing to casing-tubing annulus above the packer, J; q F is the heat flow from surrounding formation to casing-tubing annulus above the packer, J.
Based on the heat transfer theory, the heat balance for fluid flowing in tubing and in the casing-tubing annulus above the packer is respectively given by: where c w is the water specific heat capacity, J/(kg.K); w inj is the mass of the reversed fluid in the casing-tubing annulus above the packer, kg; T fh is the fluid temperature in the tubing above the packer, K; T fc is the fluid temperature in the casing-tubing annulus above the packer, K.
Heat flow, q F , from the surrounding formation to the casing-tubing annulus above the packer is given by: Heat flow, q ta , from the tubing to the casing-tubing annulus above the packer is given by: where r c is the casing radius, m; r t is the tubing radius, m; T wb is the temperature at wellbore/formation interface, K; U a is the overall heat-transfer coefficient of casing-tubing annulus above the packer, which depends on the resistances to heat flow through annular fluid, insulation material on the casing surface, casing metal, and cement, W/(m K); U t is the overall heat-transfer coefficient of tubing above the packer, which depends on the resistances to heat flow through tubing fluid, tubing metal, and insulation material on the tubing surface, W/(m K).Simplifying these equations based on the assumptions of incompressible and single-phase fluids, the following equations can be obtained: q fh (z) = q fh (z + dz) − q ta (18)  q fc (z) = q fc (z + dz) + q ta + q F (19) q fh (z + dz) − q fh (z) = c w w injbp T fh (z + dz) − T fh (z) The boundary conditions could be found that the outlet temperature of the coaxial borehole heat exchanger section is equal to the inlet temperature of the thermoelectric power generation section, and the inlet temperature of the casing-tubing annulus above the packer is equal to the temperature of the injected fluid.Then, where T fhbp (L c ) is the outlet temperature of the coaxial borehole heat exchanger section, K; T inj is the temperature of the injected liquid, K.
By applying boundary conditions, the temperature distributions along tubing and casing-tubing annulus can be solved and expressed as, In Eqs. ( 27) and ( 28), C, D, E,ξ , 1 , 2 , m and n are all constants, given as follows.

Electrical power generation
The thermoelectric modules are attached tightly on the outer surface of the tubing, and their geometric configurations are in their optimum state.When the flowing temperature in the tubing and casing-tubing annulus is known, then the temperature on the hot and cold surfaces of TEGs can be deduced from heat transfer theory 38 and expressed as follows where T mh is the temperature on the hot surface of TEG, K; T mc is the temperature on the cold surface of TEG, K; r pn is the radius after the thermoelectric modules are attached on the external surface of the tubing, m; h t is the convective heat-transfer coefficient between the produced fluid and the tube, W/(m K); h c is the convective heat-transfer coefficient between the injected fluid and the case, W/(m K).
According to the principle of the TEG, the produced voltage depends on the temperature difference and Seebeck coefficient, as given by [44][45][46] : where e is the voltage of a thermoelectric element, V.
For an element of length ( dz ), the temperature along each side of TEG stays constant.Then the total voltage, E , can be integrated along the wellbore and given by: where n φ and n x are the number of thermoelectric pairs in a circumferential circulation and the number density of thermoelectric pairs in the axial direction, respectively, dimensionless.
When the thermoelectric module is applied to a certain temperature gradient, electric power is produced, which is defined as, where R L is the external electric resistance, Ω; I is the current flowing through the circuit, A.
The current, I , is given as, where R is the internal electric resistance of TEG, Ω.By Combining Equations.( 33) with (34), then, where P is the output power, W.
The maximum power output is obtained when the external electric resistance is equal to the internal electric resistance of the TEG [47][48][49][50] , this gives, where P max is the maximum output power, W.
The power generated by a downhole thermoelectric power generation system is the sum of the powers generated by all of the segments 51 .The output power of the ith segment with a length of L i is: where L i is the length of ith segment, m.
Then, the total output power in a well is given by: where n teg is the number of the thermoelectric generator segments.By considering the Seebeck effect, the thermal power input to the hot side in the ith segment, Q Hi , and the thermal power output from the cold side in the ith segment, Q Ci , are separately given by 48,52 , where K is the thermal conductance of TEG, W/(m K).
The efficiency of TEG can be defined as,

Required pumping power
In our proposed design, a surface pump is used to inject and circulate the cold fluid in the injection pipe, casingtubing annulus, coaxial borehole heat exchanger, and tubing.To attain the required pumping power, the pressure losses in the injection pipe, casing-tubing annulus, coaxial borehole heat exchanger, and tubing should be determined.According to single-phase flow theory, the pressure losses in flow conduits above the packer can be calculated by: (33) where P ut is the pressure loss in tubing above the packer, Pa; P ua is the pressure loss in casing-tubing annulus above the packer, Pa; P dip is the pressure loss in injection pipe, Pa; ρ is the fluid density, kg/m 3 ; g is the gravita- tional acceleration, m/s 2 ; z is the step in length of pipe, m; P fut is the pressure loss due to friction in tubing, Pa; P fua is the pressure loss due to friction in casing-tubing annulus, Pa; P fdip is the pressure loss due to friction in injection pipe, Pa.Friction losses could be calculated using the universal relation, where is the friction factor, dimensionless; D is the diameter of the conduit, m; v is the velocity of the fluid, m/s.The friction factor can be calculated by multiple methods 53,54 .
The pressure losses in coaxial borehole heat exchanger can also be calculated according to Eqs. ( 42)- (44).
With the known pressure differences, the required pumping power can be obtained.
where, P pump is the required pumping power, W; Q inj is the injection rate at surface, m 3 /s; P dc is the pressure loss in the downward coaxial borehole heat exchanger (casing-tubing annulus below the packer), Pa; P uc is the pressure loss in the upward coaxial borehole heat exchanger (tubing below the packer), Pa; ω is the ratio of the flow rate entering the casing-tubing annulus above the packer to the total injection rate.
Assuming the pumping power is supplied by the produced power from the same well, then the net power is given by: where, P net is the net power, W.

Parameters for case studies
In this case study, a downhole thermoelectric power generation system will be applied to a liquid-dominated geothermal well.The top depth of the production zone in the geothermal well is 3000 m, and the thickness of the production zone is 500 m.It was completed with a 9-5/8″ casing to the bottom.A packer was set right above the production zone with 3-1/2″ tubing connected to the surface.As designed, an injection pipe with an inner diameter of 1-5/8″ is run through the casing-tubing annulus down to the depth of 3000 m, and just penetrating the packer.The daily injection rate is 500 m 3 , and the fluid diversion ratio between the fluid entering the casingtubing annulus above and below the packer is 1.The casing above the packer is coated with an insulation material with a thermal conductivity of 0.06 W/(m K), while the casing below the packer has a high heat conductivity with a thermal conductivity of 43.25 W/(m K) and is well cemented with formation.The tubing is also coated with an insulation material with a thermal conductivity of 0.06 W/(m K), while the injection pipe is perfectly insulated.The injected fluid will reverse flow upwards in tubing at 3500 m.Segmented thermometric generators are connected in series and fully mounted on the outer surface of the tubing above the packer.The data used in this study are summarized in Table 1.www.nature.com/scientificreports/For downhole thermoelectric modules, Bi 2 Te 3 -based materials are selected as the semiconductor due to their commercial availability, high performance, and proven engineering applications 19 .TEG parameters are shown in Table 2.The length of each segment of TEG is assumed to be the same as the tubing length for convenient assembly and running into the downhole.

Temperature distribution
The temperature distribution in the tubing, casing-tubing annulus, and hot and cold sides of TEG are presented in Fig. 4. As the fluid diverted from the downhole fluid diverter flows through the packer and enters the casingtubing annulus, it is heated to about 123.5 °C at the well bottom.While it reverses and flows upwards in the tubing, it releases heat to the "cold" fluid in casing-tubing annulus below the packer, and the temperature of the reversed fluid at the packer drops to 120.1 °C.When the heated fluid continuously flows upwards in tubing, heat is transferred to TEG and, The reversed fluid in casing-tubing annulus above the packer, The wellhead temperature of the reversed fluid in tubing is high up to 97.0 °C, which is hot enough to be further used on the ground.
As the fluid flows upwards in the casing-tubing annulus above the packer, it is heated up by 43.5 °C due to heat conducting from the surrounding formation and transferring from TEG and tubing.The temperature differences among the reversed fluid in tubing and the hot side of TEG, the reversed fluid in casing-tubing annulus, and the cold side of TEG are relatively small.However, the temperature differences across both sides of TEG vary from 96.3 °C at the packer to 32.2 °C at the wellhead.Such temperature differences are large enough to produce Table 2. Thermoelectric properties and parameters in this Case Study.www.nature.com/scientificreports/electric power 24 .The continuous injection of cold fluid not only maintains a lower temperature environment in the cold side of thermoelectric module but also provides heated fluid for direct use on the ground.

Power generation
In this study, the net electric power from a single geothermal well is up to 228.06 kW with the proposed design.The power capacity is bit smaller in a single well, but a large benefit will be achieved when a cluster of geothermal wells is included and the reversed fluids from tubing are sent to a binary power plant as a complementary.Thermoelectric performances are listed in Table 3.

Economical evaluation
A cost-benefit assessment was performed to evaluate the economic feasibility of downhole thermoelectric power generation in a geothermal well.Here, we assume that the geothermal well has already existed; what we only need to do is retrofit the well to be suitable for power generation.Therefore, only the capital cost of the TEG system installation and the circuitry construction are considered in the cost-benefit assessment.In addition, the power generation by geothermal energy has a large environmental benefit.More than 90% of greenhouse gas emissions could be reduced if the electrical energy produced from a fossil fuel power plant is replaced by geothermal energy 55 .Therefore, the carbon tax income is included in the cost-benefit assessment.
According to the dimensions of P-N type semiconductors and the length and diameter of tubing, a total of 126,000 thermoelectric modules (each module is made up of 16 × 16 P-N type semiconductors) are needed to be mounted on the outer surface of tubing.Assuming the cost of a thermoelectric module is 35 RMB, the costs of accessories and installations are 25% of the total cost of thermoelectric modules 56 .The cost parameters are listed in Table 4.
Electricity generation from geothermal resources results in much lower greenhouse gas (GHG) emissions than those from traditional fossil fuels 57 .According to the report by the National Renewable Energy Laboratory (NREL), the median life cycle GHG emissions from enhanced geothermal systems binary, hydrothermal flash, and hydrothermal binary plants were found to be 32 g CO 2 eq/kWh, 47 g CO 2 eq/kWh, and 11.3 g CO 2 eq/kWh respectively 58 .For power generation from coal-fired plants, the median life-cycle GHG emission is estimated to be 1018 g CO 2 eq/kWh 59 .If the average median life cycle GHG emission from geothermal resources is taken as 30 g CO 2 eq/kWh, the reduction of GHG emissions is 988 g CO 2 eq/kWh when the electricity generated from a coal-fired plant is replaced by a geothermal power plant.It is assumed that a carbon tax will be imposed on power plants in the future.Supposing carbon tax is 20-200 RMB/t CO 2 59 , then an additional income of a geothermal power plant will increase by 0.0198-0.1976RMB/kWh.The parameters used for cost-benefit analysis are shown in Table 5, and the results for different carbon tax scenarios are presented in Table 6.
It can be seen that the investment could be returned in 6 to 8 years, and the rate of return is greater than 16% for different carbon tax scenarios.Based on the above calculation, downhole thermoelectric power generation with a coaxial borehole heat exchanger is feasible and economically competitive.

Flow rate analysis
Based on the proposed configuration and taking the fluid diversion ratio as 1, the power and efficiency of the system as a function of injection rate were calculated and shown in Fig. 5. From this figure, the required pumping www.nature.com/scientificreports/power has an approximate exponential relationship with the injection rate.The increased injection rate results in an increase of friction loss in the injection pipe, and a larger pumping power is required to ensure fluid circulated in the system.Meanwhile, the increased flow velocity leads to an increase of Reynolds number and Prandtl number, which accelerate the heat transfer between the reversed fluid in the tubing and the hot side of TEG and between the reversed fluid in the casing-tubing annulus and the cold side of TEG.The enhanced heat transfer improves the temperature differences across both sides of TEG and results in a gradual increase in the power produced.But the increments of produced power are slower than the required pumping power.Therefore, the net power shows an increasing and then decreasing change with the increasing injection rate.The maximum net power is achieved around the injection rate of 600 m 3 /d.The power efficiency shows the same variation as the produced power.Figure 6 shows the power and efficiency of the system as a function of the fluid diversion ratio at an injection rate of 500 m 3 /d.With the increasing fluid diversion ratio, the reversed fluid in the casing-tubing annulus  increases and the reversed fluid in the tubing decreases.More fluids flowing in the casing-tubing annulus are beneficial to keep the cold side of TEG at a lower temperature, but the potential of lowering the temperature on the cold side of TEG by increasing the flow rate in the casing-tubing annulus may be limited.Meanwhile, the decreased fluid flowing into the coaxial borehole heat exchanger will result in less heat harvesting from the surrounding formation and lower the fluid temperature in the tubing at the packer.Therefore, the temperature difference across both sides of the TEGs shows an increasing and then decreasing change with increasing fluid diversion ratio.Thus, the produced power begins to decrease gradually after the fluid diversion ratio is higher than a certain value.The required pumping power shows a smaller change, which indicates that the friction loss may be mainly consumed in the injection pipe.The thermal-to-electricity conversion efficiency has the same variation as the produced power and the net power.From the results, the reasonable fluid diversion ratio is between 1.5 and 2.

Flow conduit analysis
The size of the flow conduit is an important factor that affects the flow velocity of the fluid, the heat transfer process, and friction losses.www.nature.com/scientificreports/tubing-casing configurations.For a constant tubing size,a smaller casing will have a smaller casing-tubing annulus, which will result in a faster flow velocity of the reversed fluid in it.Thus, more heat will be taken away from the cold side of TEGs.This is helpful to keep the cold side of TEGs in a lower temperature condition and further increase the temperature difference across both sides of TEGs (Fig. 9), which will produce more power with the fixed number of thermoelectric modules.It can be seen that the power produced decreases with increased casing size.With the increasing casing-tubing annulus, the changes in flow velocity and the fiction losses become slower and smaller, and the required pumping power becomes stable after a gradual decrease.Therefore, the net power shows an increasing and then decreasing variation with increased casing size.For a constant casing size, smaller tubing will result in a faster flow velocity in the tubing.This makes the reversed fluid in the tubing being transfer more heat to the hot side of TEG.It is helpful to keep the hot side of TEG at a higher temperature and further a larger temperature difference across both sides of TEGs (Fig. 10

Figure 2 .
Figure 2. Schematic of downhole thermoelectric power generation design with a coaxial borehole heat exchanger in a geothermal well.

Figure 3 .
Figure 3. Schematic of heat balance for tubular and formation.

Figure 4 .
Figure 4. Temperature distributions in tubing, casing-tubing annulus and both sides of TEG.

Figure 5 .
Figure 5. Powers and efficiency change with injection rate.

Figures 7 and 8 FluidFigure 6 .
Figure 6.Powers and efficiency change with fluid diversion ratio at a injection rate of 500 m 3 /d.

Table 1 .
46) P net = P t − P pump Parameters for Case Study of Downhole Thermoelectric Power Generation.

Table 3 .
Thermoelectric Performances in Case Study.

Table 4 .
Capital cost of a downhole thermoelectric power generation system.

Table 5 .
Parameters used for cost-benefit analysis.

Table 6 .
Results of cost-benefit analysis for different carbon tax scenarios. ),